302 ANALYTICAL MECHANICS a as the limits of x and the corresponding values of * as the limits of t we have ? -r, 1 a — f = 4 P 2 a 4 27r2a2m Cl . 227r, ,. p-js-)>**-¥** 6 -JJS. (IX) PROBLEMS. 1. A particle which describes a simple harmonic motion has a period of 5 sec. and an amplitude of 30 cm. Find its maximum velocity and its maximum acceleration. 2. When a load of mass m is suspended from a helical spring of length L and of negligible mass an extension equal to D is produced. The load is pulled down through a distance a from its position of equilibrium and then set free. Find the period and the amplitude of the vibration. Hooke/s law holds true. 3. Within the earth the gravitational attraction varies as the distance from the center. Suppose there were a straight shaft from pole to pole, with no resisting medium in it. What would be the period of oscillation of a body dropped into the shaft? Suppose the earth to be a sphere with a radius of 4000 miles. 4. In the preceding problem find the velocity with which the body would pass the center of the earth. 6. A particle describes a circle with constant speed. Show that the projection of the particle upon a straight line describes a simple harmonic motion. 6. The pan of a helical spring balance is lowered 2 inches when a weight of 5 pounds is placed on it. Find the period of vibration of the balance with the weight on. * See footnote p. 142.